Vol. 29, No. 1, 1969

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On indecomposable injectives over artinian rings

Kent Ralph Fuller

Vol. 29 (1969), No. 1, 115–135

For each primitive idempotent f in a left artinian ring R, let T(Rf) denote the unique simple factor of Rf and let E(T(Rf)) denote its injective hull. Then, identifying a module with its isomorphism class,

fR ← → E(T (Rf ))

provides a one-to-one correspondence between the finite sets of isomorphism classes of indecomposable injective left R-modules and indecomposable projective right R-modules. The importance of this correspondence is illustrated by the Nagao-Nakayama result that if R is a finite-dimensional algebra over a field K then, under the duality ( ) = HomK(_,K) between the categories of finitely generated left and right R-modules, one has

(fR )∗ ∼= E(T (Rf )).

Thus the structure of an indecomposable injective module E(T(Rf)) over such an algebra R is completely determined by the indecomposable (projective) direct summand fR of RR. However, in the more general case when R is a left artinian ring, very little is known about the structure of these indecomposable injective modules. In this paper we attempt to shed some light on the problem by showing that a large part of the above mentioned duality can be carried over to fR and E(T(Rf)) over a ring R with minimum condition on left ideals.

Mathematical Subject Classification
Primary: 16.50
Received: 5 June 1968
Published: 1 April 1969
Kent Ralph Fuller